In this paper we investigate the variety of idempotent commutative groupoids. In particular, we improve the results of Grätzer and Padmanabhan on the number of essentially n-ary polynomials in idempotent commutative groupoids. They have shown that if an idempotent commutative groupoid (G,•) is different from a semilattice, thenfor all n. Moreover, they have proved that the equality is achieved if and only if (G,•) is polynomially equivalent to an affine space over GF(3).We prove that if (G,•)...
G. Grätzer and A. Kisielewicz devoted one section of their survey paper concerning -sequences and free spectra of algebras to the topic “Small idempotent clones” (see Section 6 of [18]). Many authors, e.g., [8], [14, 15], [22], [25] and [29, 30] were interested in -sequences of idempotent algebras with small rates of growth. In this paper we continue this topic and characterize all idempotent groupoids with (see Section 7). Such groupoids appear in many papers see, e.g. [1], [4], [21], [26,...
In [7] and [8], two sets of regular identities without finite proper models were introduced. In this paper we show that deleting one identity from any of these sets, we obtain a set of regular identities whose models include all affine spaces over GF(p) for prime numbers p ≥ 5. Moreover, we prove that this set characterizes affine spaces over GF(5) in the sense that each proper model of these regular identities has at least 13 ternary term functions and the number 13 is attained if and only if the...
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